12 1. PRELIMINARIES
where
gradf(x) =
∂f
∂xji
n
i,j=1
.
Thus the standard Poisson-Lie bracket becomes
(1.25)
{f1,f2}SLn (x) =
1
2
(R(gradf1(x) x), gradf2(x) x)−
1
2
(R(x gradf1(x)),x gradf2(x)),
where the action of the R-matrix (1.23) on ξ = (ξij)i,j=1 n sln is given by
R(ξ) = (sign(j i)ξij)i,j=1
n
.
Substituting into (1.25) coordinate functions xij, xkl, we obtain
{xij,xkl}SLn =
1
2
(sign(k i) + sign(l j)) xilxkj.
In particular, the standard Poisson-Lie structure on
SL2 =
a b
c d
: ad bc = 1
is described by the relations
{a, b}SL2 =
1
2
ab, {a, c}SL2 =
1
2
ac, {a, d}SL2 = bc,
{c, d}SL2 =
1
2
cd, {b, d}SL2 =
1
2
bd, {b, c}SL2 = 0.
Note that the Poisson bracket induced by {·, ·}SL2 on upper and lower Borel
subgroups of SL2
B+ =
p q
0
p−1
, B− =
p 0
q
p−1
has an especially simple form:
{p, q} =
1
2
pq.
The example of SL2 is instrumental in an alternative characterization of the
standard Poisson-Lie structure on G. Fix i [1,l], l = rank g, and consider an
annihilator of the subalgebra
gi
= span{ei,e−i,hi} with respect to the Killing
form:
(gi)⊥
= hi


(
⊕α∈Φ\{α±i}gα
)
,
where hi

is the orthogonal complement of hi in h with respect to the restriction
of the Killing form. Though
(gi)⊥
is not even a subalgebra in g, it is an ideal in
g∗
equipped with the Lie bracket (1.24). It follows that for every i, the image of
the embedding ρi : SL2 G defined in (1.7) is a Poisson-Lie subgroup of G. The
restriction {·, ·}ρi(SL2) of the Sklyanin bracket (1.22), (1.23) to ρi(SL2) involves
only the restriction of the standard R-matrix to
gi
= ρi(sl2). It is not difficult to
check then, that
{·, ·}ρi(SL2) = (αi,αi){·, ·}SL2 .
Thus the standard Poisson-Lie structure is characterized by the condition that for
every i [1,l], the map
(1.26) ρi :
(
SL2, (αi,αi){·, ·}SL2
)

(
G, {·, ·}G
)
is Poisson.
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